| Title:
|
A semilattice of varieties of completely regular semigroups (English) |
| Author:
|
Petrich, Mario |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
1-14 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Completely regular semigroups are unions of their (maximal) subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $\mathcal L(\mathcal C\mathcal R)$. \endgraf We construct a 60-element $\cap $-subsemilattice and a 38-element sublattice of $\mathcal L(\mathcal C\mathcal R)$. The bulk of the paper consists in establishing the necessary joins for which it uses Polák's theorem. (English) |
| Keyword:
|
completely regular semigroup |
| Keyword:
|
lattice |
| Keyword:
|
variety |
| Keyword:
|
$\cap $-subsemilattice |
| MSC:
|
20M07 |
| idZBL:
|
07217175 |
| idMR:
|
MR4088688 |
| DOI:
|
10.21136/MB.2018.0112-17 |
| . |
| Date available:
|
2020-03-12T08:17:22Z |
| Last updated:
|
2020-11-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148057 |
| . |
| Reference:
|
[1] Hall, T. E., Jones, P. R.: On the lattice of varieties of bands of groups.Pac. J. Math. 91 (1980), 327-337. Zbl 0419.20043, MR 0615681, 10.2140/pjm.1980.91.327 |
| Reference:
|
[2] Jones, P. R.: Mal'cev products of varieties of completely regular semigroups.J. Aust. Math. Soc., Ser. A 42 (1987), 227-246. Zbl 0613.20038, MR 0869748, 10.1017/S1446788700028226 |
| Reference:
|
[3] Liu, G., Zhang, J.: A problem on central cryptogroups.Semigroup Forum 73 (2006), 261-266. Zbl 1114.20034, MR 2280823, 10.1007/s00233-006-0619-0 |
| Reference:
|
[4] Pastijn, F.: The lattice of completely regular semigroup varieties.J. Aust. Math. Soc., Ser. A 49 (1990), 24-42. Zbl 0706.20042, MR 1054080, 10.1017/S1446788700030214 |
| Reference:
|
[5] Petrich, M.: Varieties of orthodox bands of groups.Pac. J. Math. 58 (1975), 209-217. Zbl 0317.20042, MR 0382522, 10.2140/pjm.1975.58.209 |
| Reference:
|
[6] Petrich, M.: On the varieties of completely regular semigroups.Semigroup Forum 25 (1982), 153-169. Zbl 0502.20034, MR 0663176, 10.1007/BF02573595 |
| Reference:
|
[7] Petrich, M.: A lattice of varieties of completely regular semigroups.Commun. Algebra 42 (2014), 1397-1413. Zbl 1302.20059, MR 3169638, 10.1080/00927872.2012.667181 |
| Reference:
|
[8] Petrich, M.: Certain relations on a lattice of varieties of completely regular semigroups.Commun. Algebra 43 (2015), 4080-4096. Zbl 1339.20053, MR 3366561, 10.1080/00927872.2014.907412 |
| Reference:
|
[9] Petrich, M.: Some relations on a semilattice of varieties of completely regular semigroups.Semigroup Forum 93 (2016), 607-628. Zbl 06688592, MR 3572420, 10.1007/s00233-016-9817-6 |
| Reference:
|
[10] Petrich, M., Reilly, N. R.: Semigroups generated by certain operators on varieties of completely regular semigroups.Pac. J. Math. 132 (1988), 151-175. Zbl 0598.20061, MR 0929587, 10.2140/pjm.1988.132.151 |
| Reference:
|
[11] Petrich, M., Reilly, N. R.: Operators related to $E$-disjunctive and fundamental completely regular semigroups.J. Algebra 134 (1990), 1-27. Zbl 0706.20043, MR 1068411, 10.1016/0021-8693(90)90207-5 |
| Reference:
|
[12] Petrich, M., Reilly, N. R.: Operators related to idempotent generated and monoid completely regular semigroups.J. Aust. Math. Soc., Ser. A 49 (1990), 1-23. Zbl 0708.20019, MR 1054079, 10.1017/s1446788700030202 |
| Reference:
|
[13] Petrich, M., Reilly, N. R.: Completely Regular Semigroups.Canadian Mathematical Society Series of Monographs and Advanced Texts 23. Wiley, Chichester (1999). Zbl 0967.20034, MR 1684919 |
| Reference:
|
[14] Polák, L.: On varieties of completely regular semigroups. I.Semigroup Forum 32 (1985), 97-123. Zbl 0564.20034, MR 0803483, 10.1007/BF02575527 |
| Reference:
|
[15] Polák, L.: On varieties of completely regular semigroups. II.Semigroup Forum 36 (1987), 253-284. Zbl 0638.20032, MR 0916425, 10.1007/BF02575021 |
| Reference:
|
[16] Reilly, N. R.: Varieties of completely regular semigroups.J. Aust. Math. Soc., Ser. A 38 (1985), 372-393. Zbl 0572.20040, MR 0779201, 10.1017/S144678870002365X |
| Reference:
|
[17] Trotter, P. G.: Subdirect decompositions of the lattice of varieties of completely regular semigroups.Bull. Aust. Math. Soc. 39 (1989), 343-351. Zbl 0661.20039, MR 0995132, 10.1017/S0004972700003269 |
| . |
